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taloc
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Homework Statement
I am tasked with solving the QHO with a sinusoidal perturbing potential of the form VoSIN(BX). I need to find the ground state energy as well as the ground state eigenket |g>.
Homework Equations
[tex]H_{o} = \frac{P^{2}}{2m} + \frac{1}{2}m \varpi^{2} [/tex]
[tex]H = H_{o} + Asin(BX)[/tex]
[tex]E^{(o)}_{n}=\hbar\varpi(n+\frac{1}{2})[/tex] Which is the unperturbed energy
The Attempt at a Solution
My first stab at this problem involved performing a Taylor expansion of the potential and the rewriting of the X operator in terms of the creation and annihilation operators:
[tex] X = \sqrt{\frac{\hbar}{2m\varpi}}(a^{\dagger}+a)[/tex]
This process was not very rewarding. I found myself with no method for determining when to terminate the expansion.
An alternative approach would be to rewrite the potential as:
[tex]Asin(BX) = \frac{A}{2i}(e^{iBX}-e^{-iBX})[/tex]
My concern with this method is that the |n> kets used in the QHO are not eigenkets of X and therefore do not play nicely with the exponentials. Do I need to perform a change of basis? Essentially creating a new set of kets composed of a linear combination of the |n> kets? Any advice would be greatly appreciated!
Cheers!
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